In terms of the leading soft-photon operator \(S_a(\vec{w})\) , Weinberg's soft photon theorem [1]}, [2]} can be written as,
\({\begin{array}{c}{\lbrace y_i,\vec{x}_i,Q_i\rbrace ,out} S_a(\vec{w}) {\lbrace y_j,\vec{x}_j,Q_j\rbrace ,in} \nonumber \\= \bigg [ \sum _k \eta _k Q_k \frac{p(y_k,\vec{x}_k) \cdot \epsilon _a(\vec{w})}{p(y_k,\vec{x}_k)\cdot \hat{q}(\vec{w})} \bigg ] {\lbrace y_i,\vec{x}_i,Q_i\rbrace ,out}{\lbrace y_j,\vec{x}_j,Q_j\rbrace ,in}\end{array}}\) 
